Networks Part II Samu Varjonen Ashwin Rao
Models of Complex Networks to Model Overlay Networks Milgram's Experiment Duncan Watts Random Rewiring Model Scale Free Networks (Power-Law Networks) Preferential attachment Evolving Copying Model (Copying Generative Model) Navigation in Small World Complex Networks Overlay Networks P2P
Outline for this lecture Error and Attack Tolerance of Complex Networks Navigation in Complex Networks Mathematics and the Internet: A Source of Enormous Confusion and Great Potential Summary on Modeling Overlay Networks
Scale-Free Model for AS-Graph High Degree Nodes (Hubs) AS Topology of skitter dataset parsed by SNAP team http://snap.stanford.edu/data/as-skitter.html
Importance of Hubs (Random Graph) Albert, Réka, et al. "Error and attack tolerance of complex networks." nature 406, no. 6794 (2000): 378-382.
Error vs Attack Error (Node Failure) random node fails (malfunction) Attack Selected node with a given property is made to fail Which nodes would you target if you knew the network is a scale-free network? Nodes with the highest degree
Impact of Errors and Attacks (Graph Diameter) Graph Diameter Fraction of nodes removed Albert, Réka, et al. "Error and attack tolerance of complex networks." nature 406, no. 6794 (2000): 378-382.
Impact of Errors and Attacks (Size of Largest Cluster) S: Fraction of nodes in largest cluster <s>: average size of isolated clusters Relative <s> and S Fraction of nodes removed Albert, Réka, et al. "Error and attack tolerance of complex networks." nature 406, no. 6794 (2000): 378-382.
Network Response to Attacks and Failures f=0 f 0.05 f 0.18 f 0.45 f=1 Albert, Réka, et al. "Error and attack tolerance of complex networks." nature 406, no. 6794 (2000): 378-382.
Critical Threshold (random node failures) m=1 Cohen's technique can be extended to errors (No closed form for for errors ) Cohen, Reuven et al. "Resilience of the Internet to random breakdowns." Physical review letters 85, no. 21 (2000): 4626.
Summary on Attack and Error Tolerance of Complex Networks Scale-free networks resilient to random failures but vulnerable to targetted attacks
Clustering of Nodes Constant clustering coeff. Average Degree Increases How to Generate Scale-Free Graphs with Strong Clustering Serrano, M. Angeles, Dmitri Krioukov, and Marián Boguná. "Self-similarity of complex networks and hidden metric spaces." Physical review letters 100, no. 7 (2008): 078701.
Generating Scale-Free Graphs with Strong Clustering
Path Length (Greedy Routing) Avg. Length of Greedy Routing Paths Network Size Exponent Path length grows polylogarithmically with the network size Paths shorter for smaller exponents and stronger clustering Boguna, Marian et al. "Navigability of complex networks." Nature Physics 5, no. 1 (2009): 74-80.
Greedy Routing Hidden Space as the coordinate space Hidden space is circle in this example Greedy Routing: Send to neighbor who is closer to the destination (in hidden space) Unsuccessful Paths: None of your neighbors are closer to the destination in the hidden space
P(success) decays with N Success Probability (Greedy Routing) Weak clustering Strong clustering P(success) intertwined Boguna, Marian et al. "Navigability of complex networks." Nature Physics 5, no. 1 (2009): 74-80.
Navigation in Scale Free Networks PGP BGP
Implications of Result Internet Routing Routers currently exchange signals to keep cohenrent view of network Network size increasing with time Hidden metric space eliminates the need for control signals exchanged to notify changes in network How to proceed to discover the hidden metric space Does Shortest Path imply Shortest Time to destination? What happens in case of congestion at hubs?
Scale-Free Model for AS-Graph AS Topology of skitter dataset parsed by SNAP team http://snap.stanford.edu/data/as-skitter.html
Is the Scale-Free Internet A Myth? What we have seen till now wrt to Preferential Attachment Preferential attachment results in Hubs Hubs vulnerable to coordinated attacks Why is the Internet still up and running Is the Scale-Free modeling paradigm consistent with the engineered nature of the Internet and the design constraints imposed by existing technology? Is the simplistic toy model too generic? Do the available measurements, their analysis, and their modeling efforts support the claims made by Error and Attack Tolerance paper?
Importance of Measurements Tool for measurement study for AS-measurements Traceroute Biases of traceroute Uses IPv4 Protocol What about non-ipv4 protocols like MPLS? Entry points to non-ipv4 regions can aggregate to Hubs Only reports the interfaces traversed by the packet Routers can have multiple interfaces and appear on different routes with different IP addresses
Leverage Domain Knowledge Device Constraints Finite number of interfaces on routers Finite capacity of routers Placement of High Degree Nodes What about Overlay Networks? Edge vs Core How would you deploy the network if you are a network engineer? Leverage domain knowledge to identify driving forces behind the design of high engineered systems such as the Internet
Scale Free for Gnutella? Gnutella August 2002 dataset parsed by SNAP team http://snap.stanford.edu/data/p2p-gnutella31.html
Recap of Modeling Overlay Networks Milgram's Experiment Duncan Watts Random Rewiring Model Scale-Free Networks Preferential attachment Evolving Copying Model (Copying Generative Model) Scale-Free with Strong Clustering Error and Fault Tolerance of Complex Networks Navigation (Greedy Routing) In Small World (Kleinberg's Small World) In Complex Networks (Scale-Free with Strong Clustering) Mathematics and the Internet: A Source of Enormous Confusion and Great Potential
Commonly used metrics Clustering Coefficient Diameter Degree Distribution
Methodology 1) Make observations (conduct measurement studies) 2)Build model to explain observations Choose the right level of granularity (zoom level) Strip the problem to a simple form Attempt to formulate the problem and model the system 3)Validate model Reproduce observations/measurements Explain observations 4)Revisit step 2 (and 1) to improve understanding
Important Articles Milgram, Stanley. "The small world problem." Psychology today 2.1 (1967): 60-67 Watts, Duncan and Strogatz, Steven. "Collective dynamics of small-world networks." Nature 393.6684 (1998): 440-442. Barabási, Albert-László, and Albert, Réka. "Emergence of scaling in random networks." Science 286, no. 5439 (1999): 509-512. Kleinberg, Jon. "The small-world phenomenon: An algorithmic perspective." In ACM Symposium on Theory of computing, pp. 163-170. 2000. Ravi Kumar et al. "Stochastic models for the web graph." In Annual Symposium on Foundations of Computer Science, 2000. Albert, Réka, and Barabási, Albert-László. "Statistical mechanics of complex networks." Reviews of modern physics 74.1 (2002): 47. Newman, Mark. "The structure and function of complex networks." SIAM review 45, no. 2 (2003): 167-256. Mitzenmacher, M. (2004). "A brief history of generative models for power law and lognormal distributions." Internet mathematics, 1(2), 226-251. Mark Newman. "Power laws, Pareto distributions and Zipf's law." Contemporary physics 46, no. 5 (2005): 323-351 Jure Leskovec et al. "Graphs over time: densification laws, shrinking diameters and possible explanations." In ACM SIGKDD, pp. 177-187. 2005. Boguna, Marian et al. "Navigability of complex networks." Nature Physics 5, no. 1 (2009): 74-80. W Willinger et al. Mathematics and the internet: A source of enormous confusion and great potential. In Notices of the AMS. 2009.
Sources for these slides Sasu Tarkoma Overlay and P2P Networks, 2015 Datasets from Stanford Network Analysis Project (SNAP)